A major Saskatchewan potash producer ships potash from mines in Saskatchewan to several warehouses in North America by rail. The objective is to prevent stock-out at the warehouses. The demand from the warehouses is very hard to predict. It is rarely the same every day and is often impacted by weather. In addition, the transit time is long and variable. The potash producer has to continually monitor demand and rail service patterns to avoid running out of product. Typically, there will be safety stock in the warehouses and the potash producer will also hedge its position by advancing trains when it is estimated that the busy season will start. The warehouses have limited capacity. There are situations where a train has to be held out of a site if they cannot handle it. There can be significant costs to holding a train. The other option is to divert a train to another location but that also results in additional costs. Consider a specific problem. A customer's warehouse in Florida requires continuous replenishment with potash. The warehouse has a total storage capacity of 10,000 tonnes. There is a potash producer's mine in Saskatchewan that has 50,000 tonnes of inventory of the required product. Empty cars have to be staged at the mine two days before loading. The loading for up to 100 cars takes only one day. The train with up to 100 cars gets pulled by the originating rail carrier from the mine to Chicago, where some or all of the cars get interchanged to a US rail carrier for forwarding to Florida (the other cars, if any, go to a different customer). Then, the US carrier interchanges the cars to a short line carrier that provides local rail service to the warehouse in
Florida. The rail transit from Saskatchewan to Florida takes on average 20 days (but could vary between 12 and 40 days). The short line can carry 20 cars per day to the warehouse. The warehouse can unload 20 cars per day and would typically unload the cars the same day they are delivered to the site. Each car has a capacity of 100 tonnes.

Suppose that the busy season has started and demand from the Florida warehouse is expected to be 700 tonnes per day on average for the next month or so. There are a total of 180 cars (18,000 tonnes of potash) already on the way to the Florida warehouse, shipped in two trains (9,000 tonnes each) that are expected to arrive at the warehouse in 4 days and 13 days, respectively. There will be no other shipments to the Florida warehouse during the next 28 days. There are currently 4,500 tonnes of potash in the warehouse.

a. What is the average time to replenish the warehouse with 10,000 tons of potash?

b. How many cars of potash should the potash producer dispatch to the Florida warehouse today? The potash producer would like to fill up the Florida warehouse to its limit. You may use Excel.